Model prediction of pumping performance of optimal duct designs in the function space of flow rate versus adverse pressure gradient ( Q , dp /dz ) are shown in the solid grey line. Non-optimal ducts (dashed grey lines) at small cilia-to-lumen ratio h / H and lumen diameter H generate lower pressure and flow rate, and non-optimal ducts at high h / H and H perform less efficiently while using more cilia. Here we fix the reference resistance constant to be equal to the fluid viscosity . For results using other values of , see Supplementary Fig. 2 . Biological data from Fig. 2a are superimposed using analogous symbols. Triangles represent systems with known filtration functions. Circles denote systems with known bulk transport (trans.) functions. Squares represent systems with currently unknown or disputed functions. Symbols with solid boundaries indicate perpendicular ciliation, and symbols with a central dot have longitudinal ciliation. Here we set the colour of the filtration systems to be different from those for bulk transport to show that filtration systems are separated from the transport systems in functional space. The size of the markers indicates the relative amount of ciliated area per cross section (Fig. 3f ). Operating curves (dashed pink lines) of the larvacean ciliated funnel and oesophagus carpet are highlighted. Our model predicts that most biological ciliated ducts operate close to the optimal functional limits.
